Subgroups — #31774734

Order 17344.

810 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22576208	C₂	1	5	yes	maximal, complemented
2	#22576206	C₂	1	10	yes	maximal, complemented
3	C₂×#16864655	C₂	1	5	yes	maximal, complemented
4	#41×D₂₇₁	C₂	1	10	yes	maximal, complemented
5	2₋¹⁺⁴×C₂₇₁	C₂	1	1	yes	maximal, characteristic, complemented
6	#16864655	C₂²	1	10	yes	complemented
7	#16864655	C₂²	1	20	yes	complemented
8	#16864654	C₂²	1	30	yes	complemented
9	#16864653	C₂²	1	10	yes	complemented
10	Q₈×D₂₇₁	C₂²	1	10	yes	complemented
11	C₂×D₁₀₈₄	C₂²	1	15	yes	complemented
12	D₄×D₂₇₁	C₂²	1	30	yes	complemented
13	C₂×C₄×D₂₇₁	C₂²	1	15	yes	complemented
14	#41×C₂₇₁	C₂²	1	10	yes	complemented
15	C₂×Q₈×C₂₇₁	C₂²	1	5	yes	complemented
16	C₂²×D₂₇₁	C₂³	1	15	yes	—
17	Dic₅₄₂	C₂³	1	10	yes	—
18	D₁₀₈₄	C₂³	1	30	yes	—
19	C₂²⋊D₂₇₁	C₂³	1	20	yes	—
20	C₂×Dic₂₇₁	C₂³	1	5	yes	—
21	C₄×D₂₇₁	C₂³	1	10	yes	—
22	C₄×D₂₇₁	C₂³	1	30	yes	—
23	Q₈×C₂₇₁	C₂³	1	10	yes	—
24	D₄×C₂₇₁	C₂³	1	10	yes	—
25	C₂×C₄×C₂₇₁	C₂³	1	15	yes	—
26	C₂²×C₂₇₁	C₂⁴	1	5	yes	—
27	C₂×D₂₇₁	—	4	15	no	—
28	C₂×D₂₇₁	C₂⁴	1	10	yes	—
29	C₄×C₂₇₁	C₂⁴	1	10	yes	—
30	Dic₂₇₁	C₂⁴	1	1	yes	characteristic
31	Dic₂₇₁	C₂⁴	1	5	yes	—
32	C₂×C₂₇₁	—	2	5	no	—
33	C₂×C₂₇₁	C₂⁵	1	1	yes	characteristic
34	D₂₇₁	—	2	10	no	—
35	C₂₇₁	#585	1	1	yes	characteristic, complemented, Sylow
36	#585	—	271	1	no	maximal, Sylow
37	2₋¹⁺⁴	—	271	5	no	—
38	2₋¹⁺⁴	D₂₇₁	1	1	yes	characteristic, complemented
39	H(5,𝔽₂)	—	271	10	no	—
40	C₂×#41	—	271	10	no	—
41	C₂×#41	—	271	5	no	—
42	#41	—	271	10	no	—
43	#41	—	271	20	no	—
44	#41	—	271	10	no	—
45	#41	—	271	30	no	—
46	#41	C₂×D₂₇₁	1	10	yes	complemented
47	C₂×Q₈	—	271	10	no	—
48	C₂×Q₈	C₂×D₂₇₁	1	5	yes	complemented
49	C₂×D₄	—	271	30	no	—
50	C₂×D₄	—	271	15	no	—
51	C₂²×C₄	—	271	15	no	—
52	C₂³	—	271	15	no	—
53	Q₈	—	271	10	no	—
54	Q₈	C₂²×D₂₇₁	1	10	yes	—
55	D₄	—	271	20	no	—
56	D₄	—	271	30	no	—
57	D₄	C₂²×D₂₇₁	1	10	yes	—
58	C₂×C₄	—	271	5	no	—
59	C₂×C₄	—	271	10	no	—
60	C₂×C₄	—	271	30	no	—
61	C₂×C₄	C₂²×D₂₇₁	1	15	yes	—
62	C₂²	—	271	10	no	—
63	C₂²	—	1084	15	no	—
64	C₂²	C₂³×D₂₇₁	1	5	yes	—
65	C₄	—	271	1	no	—
66	C₄	—	271	5	no	—
67	C₄	C₂³×D₂₇₁	1	10	yes	—
68	C₂	—	2	5	no	—
69	C₂	—	542	10	no	—
70	C₂	C₂⁴×D₂₇₁	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂⁴×D₂₇₁
commutator, socle, cosocle	C₂×C₂₇₁	C₂⁵
Fitting	2₋¹⁺⁴×C₂₇₁	C₂
hypercenter	2₋¹⁺⁴	D₂₇₁
hypocenter	C₂₇₁	#585
LCS	G ▷ C₂×C₂₇₁ ▷ C₂₇₁
UCS	C₁ ◁ C₂ ◁ 2₋¹⁺⁴

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. #22576208	C₂	semidirect
2. #22576206	C₂	semidirect
3. C₂×#16864655	C₂	semidirect
4. #41×D₂₇₁	C₂	semidirect
5. 2₋¹⁺⁴×C₂₇₁	C₂	semidirect
6. #16864655	C₂²	semidirect
7. #16864655	C₂²	semidirect
8. #16864654	C₂²	semidirect
9. #16864653	C₂²	semidirect
10. Q₈×D₂₇₁	C₂²	semidirect
11. C₂×D₁₀₈₄	C₂²	semidirect
12. D₄×D₂₇₁	C₂²	semidirect
13. C₂×C₄×D₂₇₁	C₂²	semidirect
14. #41×C₂₇₁	C₂²	semidirect
15. C₂×Q₈×C₂₇₁	C₂²	semidirect
35. C₂₇₁	#585	semidirect
38. 2₋¹⁺⁴	D₂₇₁	semidirect
46. #41	C₂×D₂₇₁	semidirect
48. C₂×Q₈	C₂×D₂₇₁	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#22576208	2
#22576206	2
C₂×#16864655	2
#41×D₂₇₁	2
2₋¹⁺⁴×C₂₇₁	2
#585	271